Computing Sparse Multiples of Polynomials

Abstract

We consider the problem of finding a sparse multiple of a polynomial. Given f ∈ F[x] of degree d, and a desired sparsity t, our goal is to determine if there exists a multiple h ∈ F[x] of f such that h has at most t non-zero terms, and if so, to find such an h. When F=ℚ and t is constant, we give a polynomial-time algorithm in d and the size of coefficients in h. When F is a finite field, we show that the problem is at least as hard as determining the multiplicative order of elements in an extension field of F (a problem thought to have complexity similar to that of factoring integers), and this lower bound is tight when t = 2.

The authors would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC), and MITACS.